Within the LISA context, bounds on \(\delta c_{\rm GW}\)  have previously been considered in two different settings. First, [1]} investigated testing a frequency-independent \(\delta c_{\rm GW}\)  in the LISA band with a multi-messenger observation akin to GW170817. They forecasted a resulting bound \(|\delta c_{\rm GW}| \lesssim 10^{-12}\)  in the event of a non-detection. As before, here we instead focus on candidate signatures of a frequency-dependent \(\delta c_{\rm GW}\)  and (given our findings in the LVK band above) expect to place significantly stronger bounds on such a signal. Secondly, while this paper was being completed [2]} also investigated a frequency-dependent \(c_{\rm GW}\)  in the LISA band, similarly motivated by the observation of [3]} that a frequency-dependent \(c_{\rm GW}\)  can lead to interesting phenomenology close to the LISA band and exploring its effect on waveform models and on (forecasted) parameter constraints.
In terms of \(\delta c_{\rm GW}\)  constraints, their results are driven by redshift-induced frequency dependence (e.g. that the frequency of a monochromatic wave differs at the source and at the observer in an expanding Universe).
This effect also is a generic consequence of a frequency-dependent \(c_{\rm GW}\)  and [2]} uses this to place bounds on \(|\delta c_{\rm GW}| \lesssim 10^{-4}\) . Here we show that these bounds can be improved by several orders of magnitude, more specifically to \(|\delta c_{\rm GW}| \lesssim 10^{-17}\) , by considering the effects of \(c_{\rm GW}\ne c\)  outlined in section .
